The central problem of Bayesian inference is to estimate the posterior distribution \(p(z\vert X)\)  of hidden variables \(z\) , given observations \(X\) , a likelihood model \(p(X\vert z)\) , and a prior distribution \(p(z)\) .
The Variational Inference (VI) approach
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consists in finding the best approximation in Kullback-Leibler (KL)-divergence to the posterior from within a family of tractable densities \(\mathcal {Q}\) . This is posed as the following optimization problem:
\(\operatornamewithlimits{arg\,min}_{q\in \mathcal {Q}} \left\lbrace  D^{KL}(q) \stackrel{\text{def}}{=}\int q(z)\ln \frac{q(z)}{p(z|X)} dz \right\rbrace \) 
